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<b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span>, <font color="Maroon">U2</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">h</font> being <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T1</font> holds  <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:45_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">h</font> being <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T1</font> holds  <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:45_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span> holds <br/><font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T1</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1_2"/>
<font color="Maroon">U1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">U2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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